Height growth on semisimple groups

TL;DR

This paper establishes a condition for lattice point counting on semisimple groups, leading to height asymptotics aligned with the Batyrev-Manin Conjecture for specific intrinsic heights.

Contribution

It introduces a new condition enabling asymptotic analysis of lattice points and height functions on semisimple groups, connecting geometric counting with number-theoretic conjectures.

Findings

Lattice point counting function is asymptotic to volume growth under the new condition.

Derived height asymptotics consistent with the Batyrev-Manin Conjecture.

Applicable to certain intrinsically defined heights on semisimple groups.

Abstract

A condition is given, under which a general lattice point counting function is asymptotic to the corresponding ball volume growth function. This is then used to give height asymptotics in the style of the Batyrev-Manin Conjecture for certain intrinsically defined heights on semisimple groups.